> Matheology § 238> > > Sir,> I just came across your paper on "Cantor's Theorem" that there is no> bijection from a set to its power set. I think you are right about the> set M of "non-generators" being paradoxical. [...]> I have been troubled about set theory since they told me in school> that there is a rational between every two irrationals, yet more> irrationals than rationals. It is obvious that "between every two> irrationals there is a rational" implies that there are as many> rationals as irrationals.

That would only need to hold if the numbers of each were finite, but note that between any two rationals there are also other rationals and between any two irrationals there are also other irrationals.

Not when there are more irrationals between any two rationals than rationals between any two irrationals, which is the case.

> However, I am frustrated that this could be> so hard to prove while being immediately obvious to the intuition.

Intuitions about infiniteness are frequently wrong.

> I> am also convinced that there cannot be more distinct Dedekind cuts> than distinct rational numbers.

Since it is well known, and provable, that a set has always more subsets than elements, why should that not allow more Dedekind cuts than mere members of Dedekind cuts.

> Just drawing a sketch of some Dedekind> cuts convinces me. The Dedekind cuts are 1) nonempty and 2) totally> ordered by the relation "is a proper subset of". For finite sets it is> easy to see, and prove by induction, that for such a collection of> sets there are no more sets than elements. But I do not know how to> make this a transfinite induction.

Possibly because the conclusion you are aiming to "prove" is false.

> You were right. The reason is: There are at most countably many finite> definitions like e = SUM1/n!. That is undisputed. So if there should> be uncountably many reals, most of them cannot be defined - or can> only be defined by infinite sequences.

Pi and sqrt(2) are both reals defined other than by infinite sequences.

> But that means the same as> being undefined, because none of those sequences defines a number> unless you know the last digit - which is impossible.

Thus WM here claims that neither pi nor sqrt(2) can be defined!

> So those "reals"> cannot be used in mathematics (which means communication) because they> cannot be communicated.

Now WM claims that neither pi nor sqrt(2) cannot be used in math because they cannot be communicated.

> They are not really real.

If they are not really real is WM really real? The world would be better off if he were not.

And here comes a> simple proof that the notion of uncountablility is in fact nonsense:> Construct all real numbers of the unit interval as infinite paths of> the complete infinite Binary Tree. It contains all real numbers> between 0 and 1 as infinite paths i. e. infinite sequences of bits.> [...]> The complete tree contains all infinite paths. The structure of the> Binary Tree excludes that are any two initial segments, B_k and B_(k> +1), which differ by more than one infinite path.

Actually, there are infinitely many, uncountably many, paths containing any FISON ( finite initial segment of nodes) B_k which will not contain any FISON B_(k+1) which is not sub-FISON of B_k.